Question

Solve the following differential equations : (x + cosy)dx + (-xsiny)=0

Answer 1

$$\dfrac{\partial Q}{\partial x}=-\sin y=\dfrac{\partial P}{\partial y}$$

$$u=\int (x+\cos y)dx+\varphi(y)$$

$$=\dfrac{x^2}{2}+x\cos y+\varphi(y)$$

$$\dfrac{\partial u}{\partial y}=-x\sin y+\varphi '(y)=-x\sin y$$

$$\varphi '(y)=0$$

$$\varphi(y)=-C$$

$$u=\dfrac{x^2}{2}+x\cos y-C$$

The general solution of the exact differential equation is given by

$$\dfrac{x^2}{2}+x\cos y=C$$